Problem 69 - College Algebra

Question

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. Volume is \(10 x^{3}+27 x^{2}+2 x-24,\) length is \(5 x-4\) width is \(2 x+3 .\)

Step-by-Step Solution

Answer

The height of the box is \(x + 2\).

Step 1: Identify Formula for Volume of a Box

The volume of a box is given by the formula: Volume = Length × Width × Height. We need to express the height of the box, so we can rearrange the formula to solve for Height: Height = Volume / (Length × Width).

Step 2: Setup the given expressions

We need to express the height using the given volume and the length and width expressions:- Volume = \(10x^{3} + 27x^{2} + 2x - 24\)- Length = \(5x - 4\)- Width = \(2x + 3\)The problem asks us to express the height as an algebraic expression by dividing the volume by the product of length and width.

Step 3: Calculate the Product of Length and Width

The first step is to multiply the given expressions for length and width:\[ (5x - 4)(2x + 3) \]Using the distributive property (FOIL method), we have:\[5x \cdot 2x + 5x \cdot 3 - 4 \cdot 2x - 4 \cdot 3 = 10x^2 + 15x - 8x - 12\]Simplifying, this becomes:\[ 10x^2 + 7x - 12 \]

Step 4: Divide the Volume by the Product

We need to divide the volume by the product calculated in the previous step to find the height:\[\frac{10x^{3} + 27x^{2} + 2x - 24}{10x^{2} + 7x - 12}\]We will use polynomial long division to find the quotient, which will be the height.

Step 5: Perform Polynomial Long Division

1. Divide the first term of the numerator by the first term of the denominator: \(\frac{10x^3}{10x^2} = x\).2. Multiply \(x\) by the entire divisor \((10x^2 + 7x - 12)\) and subtract from the original volume: - \(x(10x^2 + 7x - 12) = 10x^3 + 7x^2 - 12x\) - Subtract: \((10x^3 + 27x^2 + 2x - 24) - (10x^3 + 7x^2 - 12x) = 20x^2 + 14x - 24\)3. Repeat the process: - Divide \(\frac{20x^2}{10x^2} = 2\). - Multiply, subtract: - \(2(10x^2 + 7x - 12) = 20x^2 + 14x - 24\) - Subtract and you will have no remainder.Thus, the height of the box is \(x + 2\).

Step 6: Express the Height Algebraically

Using polynomial division, we found that the height of the box is expressed by the polynomial:\[x + 2\]This is the final algebraic expression for the height.

Key Concepts

Volume of a Box

The volume of a box is a measure of how much space it occupies. You might think of it as how many 'cubes' of a certain size can fill up the box. This concept is crucial in many areas, not just in math problems, but also in real-life situations where you need to know how much something can hold, like a container or a shipping box.

To calculate the volume of a box mathematically, you use the formula:

• Volume = Length × Width × Height

This formula is simple but powerful! Each factor—length, width, and height—represents one dimension of the box, and multiplying them together gives you the box's total volume. For the given exercise, the volume is expressed as a polynomial instead of a straightforward number, adding a layer of complexity because algebraic expressions provide flexible and detailed ways to describe dimensions.

Algebraic Expressions

Algebraic expressions are like sentences made with numbers, variables, and operations. They might look complex at first glance, but they are simply tools for showing variables and how they interact with each other in mathematical sentence form.

In our box problem, we have the algebraic expression for the volume:

• Volume: \( 10x^3 + 27x^2 + 2x - 24 \)

This expression is a way to articulate the relationship among length, width, and height of the box using a single variable, \(x\). It lets us understand how changing \(x\) affects the entire box's dimensions and therefore its full volume. The task was to discover how this expression can be split evenly by the combined length and width expression to unearth the enclosed height.

Distributive Property

The distributive property is a fundamental mathematical principle that allows us to simplify expressions. It's a rule stating that multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

• For example: \( a(b + c) = ab + ac \)

In the context of our box problem, we use the distributive property as part of calculating the product of length and width:

• Length: \(5x - 4\)
• Width: \(2x + 3\)

We used the distributive property to expand the multiplication of these binomials, allowing us to combine terms and simplify the expression to easily use in the division process. Understanding this principle is key to tackling polynomial division and working with algebraic expressions effectively.